The generic solutions of nonlinear, dispersive wave equations, such as the Korteweg-de Vries (KdV) and nonlinear
Schrödinger (NLS) equations, are the solitary wave, soliton for integrable equations, and the dispersive shock wave
(DSW), also termed an undular bore in fluid mechanics applications. Of these, the solitary wave (soliton) is the most
widely studied [1]}, [2]} as it is a steady solution, so that its solution is given by an ordinary differential
equation, rather than the original full, dispersive, nonlinear partial differential equation. In addition, for integrable
equations, solitons naturally arise from the point spectrum of the associated scattering problem [1]}, [2]}. In
general, a DSW is a modulated wavetrain which links two distinct states in a smooth manner, in contrast to a compressible flow
shock, which is a discontinuous jump between two flow states [1]}. It should be noted that in fluid mechanics,
bores come in two general forms, viscous bores and undular bores [6]}. A viscous bore is a steady wave form due to
a balance between viscosity and dispersion/nonlinearity, so that viscous bores dissipate energy [1]}, [6]}, [9]}.
In contrast, for a DSW, an undular bore in fluid mechanics, there is no loss and it is an unsteady, continually expanding,
modulated wave for which dispersion stops nonlinear breaking, as occurs for a gas dynamic shock. In its standard form, a
DSW is a modulated periodic wavetrain, with solitary waves at one edge and linear dispersive waves at the other
[6]}, the classic example being the KdV DSW. DSWs arise in a wide range of applications, including meteorology
[11]}, [12]}, [13]}, oceanography [14]}, [15]}, water waves [15]}, [17]}, geophysics
[18]}, [19]}, [20]}, [21]}, nonlinear optics [22]}, [23]}, [24]}, [25]}, Bose-Einstein condensates
[26]}, Fermionic fluids [27]} and colloidal media [28]}, see [6]} for a summary and
appreciation of these applications.
